✨ TL;DR
This paper establishes a mathematical duality framework for adversarial total variation, showing that adversarial training of binary classifiers can be understood through nonlocal calculus of variations. The work provides rigorous characterizations of subdifferentials using dual representations and integration by parts formulas in both metric and Euclidean spaces.
Adversarial training, a technique for making machine learning classifiers robust to small input perturbations, lacks a complete mathematical foundation. While it has been recognized that adversarial training can be reformulated as regularized risk minimization with a nonlocal total variation term, the mathematical properties of this regularizer—particularly its subdifferential structure—remain poorly understood. Understanding the subdifferential is crucial for optimization algorithms and theoretical analysis of adversarial robustness, but the nonlocal nature of the adversarial total variation makes standard calculus of variations techniques inapplicable.
The authors develop a duality theory for the adversarial total variation by deriving a dual representation analogous to classical total variation duality. They introduce nonlocal versions of gradient and divergence operators and establish an integration by parts formula that connects these operators. The framework is developed in two settings: first for continuous functions vanishing at infinity on proper metric spaces (providing generality), and second for essentially bounded functions on Euclidean domains (providing practical applicability). Using these dual representations, they characterize the subdifferential of the adversarial total variation through variational techniques.